20
VIII – Cauchy Theory
Having said this, let us show that the map h : I −→ C
0 (I) is continuous if
and only if so is the map σ : I × I −→ C. If indeed the latter is continuous, it
is so uniformly since I × I is compact (Chap. V, § 2, no 8); in particular, this
means that for all r > 0, there exists r
> 0 such that
s − s
≤ r
=⇒
σ(s, t) − σ(s
, t)
≤ r for all t ∈ I ;
but since h(s) ∈ C
0 (I) is just the path t → σ(s, t), this relation can be written
s − s
≤ r
=⇒
h(s) − h(s
)
I
≤ r .
(3.3)
Hence h is continuous. Proving the converse amounts to showing that (3) implies continuity of σ(s, t) at all points (s, t) ∈ I × I. To see this, let us start
with the inequality
σ
s
, t
− σ(s, t)
≤
σ
s
, t
− σ
s, t
+
σ
s, t
− σ(s, t)
and choose some r > 0. If |s − s
| ≤ r
, by (3), the first term on the right hand
side is ≤ r for all t
; but as the function t → σ(s, t) is continuous for given s, for
given (s, t), ≤ r if |t − t
|, the second term on the right hand side is sufficiently
small, qed.
Let us now consider an open subset G of C and let C
0 (I, G) be the subset of
in C
0 (I) consisting of paths I −→ G ; it is open in C
0 (I) since if μ ∈ C
0 (I, G),
the image μ(I) is a compact subset of G whose distance R to the border of G
is strictly positive;
18 It is then obvious that any path ν : I −→ C such that
μ − νI < R remains a path in G that is in fact homotopic to μ as the line
segment
σ(s, t) = (1 − s)μ(t) + sν(t)
connects μ and ν in C
0 (I). On the other hand, it is obvious that for given
a, b ∈ G, the set C
0
a,b (G) of continuous points I −→ G connecting a to b in G
is a closed subset of the open set C
0 (I, G). The same holds for the set of closed
paths in G.
In conclusion, two paths with given endpoints a and b in G are fixedendpoint homotopic if and only if they can be connected by a continuous path
in the space C
0
a,b (G) of all such paths. A similar result holds for a homotopy
of closed paths.
(ii) Differentiation with respect to a path. A norm can be defined in the
vector space C
1/2 (I) of admissible paths I −→ C by setting
μ = μ I + μ
I ;
equipped with it, C
1/2 (I) is complete. Indeed, if (μ n ) is a Cauchy sequence,
the functions μ n (t) and μ
n (t) converge uniformly to some limits μ and ν that
are respectively continuous and regulated; the relation
18 Let F be this border; the function d(z, F ) is continuous on the compact set μ(I),
and so reaches its minimum at some point a ∈ μ(I); were this minimum zero,
there would be a sequence of points of F converging to a. This would imply that
a ∈ F since F is closed, a contradiction.
VIII – Cauchy Theory
Having said this, let us show that the map h : I −→ C
0 (I) is continuous if
and only if so is the map σ : I × I −→ C. If indeed the latter is continuous, it
is so uniformly since I × I is compact (Chap. V, § 2, no 8); in particular, this
means that for all r > 0, there exists r
> 0 such that
s − s
≤ r
=⇒
σ(s, t) − σ(s
, t)
≤ r for all t ∈ I ;
but since h(s) ∈ C
0 (I) is just the path t → σ(s, t), this relation can be written
s − s
≤ r
=⇒
h(s) − h(s
)
I
≤ r .
(3.3)
Hence h is continuous. Proving the converse amounts to showing that (3) implies continuity of σ(s, t) at all points (s, t) ∈ I × I. To see this, let us start
with the inequality
σ
s
, t
− σ(s, t)
≤
σ
s
, t
− σ
s, t
+
σ
s, t
− σ(s, t)
and choose some r > 0. If |s − s
| ≤ r
, by (3), the first term on the right hand
side is ≤ r for all t
; but as the function t → σ(s, t) is continuous for given s, for
given (s, t), ≤ r if |t − t
|, the second term on the right hand side is sufficiently
small, qed.
Let us now consider an open subset G of C and let C
0 (I, G) be the subset of
in C
0 (I) consisting of paths I −→ G ; it is open in C
0 (I) since if μ ∈ C
0 (I, G),
the image μ(I) is a compact subset of G whose distance R to the border of G
is strictly positive;
18 It is then obvious that any path ν : I −→ C such that
μ − νI < R remains a path in G that is in fact homotopic to μ as the line
segment
σ(s, t) = (1 − s)μ(t) + sν(t)
connects μ and ν in C
0 (I). On the other hand, it is obvious that for given
a, b ∈ G, the set C
0
a,b (G) of continuous points I −→ G connecting a to b in G
is a closed subset of the open set C
0 (I, G). The same holds for the set of closed
paths in G.
In conclusion, two paths with given endpoints a and b in G are fixedendpoint homotopic if and only if they can be connected by a continuous path
in the space C
0
a,b (G) of all such paths. A similar result holds for a homotopy
of closed paths.
(ii) Differentiation with respect to a path. A norm can be defined in the
vector space C
1/2 (I) of admissible paths I −→ C by setting
μ = μ I + μ
I ;
equipped with it, C
1/2 (I) is complete. Indeed, if (μ n ) is a Cauchy sequence,
the functions μ n (t) and μ
n (t) converge uniformly to some limits μ and ν that
are respectively continuous and regulated; the relation
18 Let F be this border; the function d(z, F ) is continuous on the compact set μ(I),
and so reaches its minimum at some point a ∈ μ(I); were this minimum zero,
there would be a sequence of points of F converging to a. This would imply that
a ∈ F since F is closed, a contradiction.